Refraction Index

By Bento, Luis San Miguel
Posted on 2018-10-16    Last edited on 2018-10-20

When a light beam pass from air into a liquid, light velocity decreases. This provokes a

 phenomenon design as light refraction, that is quantified by the quotient:

nR   = c / v                                                                   (1)


with nR    being the refractive index and c and v the light velocity in vacuum and liquid, respectively (Mansfield M. et al.).

In sugar industry, this index is used to calculate the density and dry matter of sugar solutions and to control solutions supersaturation in crystallization. Determination of refractive index is done using refractometers (Abbe refractometer, for example). Tables with nR values are available in literature (ICUMSA-Method SPS3;2000).

Here we present a methodology to calculate refraction index of light at different wavelengths and with liquids at different temperatures, based on a material particles model developed by the authors (see Entry: Material particles – a model)

Depending on subject we use the following units for space and time ( L and T): 1 ε = 2,02025E-36 m (L)  and  1 Ω = 1 ε/c (T) (c being the light velocity in vacuum) , for Planck dimension scale; and 1 pm 1(L) and 1 τ (T), for Lab scale, with 1 t , representing the time that light takes to dislocate 1 pm. This means that maximum velocity, c = 1 pm. τ-1.

For calculations, we represent light wavelengths in pm units (1 pm = 10-12 m). However, in the text we may refer these values in nm, nanometres. 

In a liquid (we will refer, for example, water) molecules are continuously in movement, keeping the same distance between molecules centre, at a given temperature we refer as do ε.

Water molecules are permanently in decay process with formation of X Beams (photon beams) and Z Beams (null particles, e(0), beams).

Each water molecule maintain contact with other water molecules though these X Beams. Pairs of these beams are formed between molecules maintaining their velocity (inertia).

At Figure 1 we represent two material particles molecules and a light beam of λ pm wavelength.


Figure 1 – Two material particles and a light beam  


When a light beam, with λ pm of wavelength cross the liquid (water, in this example) the photons of light can inter-act with null particles from X Beams produced by water molecules. This interference can decrease the photon “migration” through the water. Final light “velocity” will decrease.

We consider a standard distance ds ε, as the distance light travel in vacuum, during a standard time we consider for our measurements.

The value Δo  ε, will be the difference of light path in water and in vacuum, during the standard time.

The refractive index nR   can be presented by:


nR   = ds / ( ds  – Δo)                                                     (2)


Considering Δ =  Δo/ ds, equation (2) can be represented as:


                                                Δ  =  1  – 1/nR                                                              (3)


To calculate refractive indexes of liquids, we calculate D value through the following equation:


                                    Δ   =  Kp. Φλ.Φd .δ.(m/m0)*                  pm                                (4)




Kp – being the probability to have the necessary conditions in order to have a photon


Φλ - is the activity of light wavelength radiation, per 1 pm (capacity to cause refraction);

Φd  - is the activity of axis D connecting two liquid molecules, per 1 pm;

δ -  in pm units, is the decrease of light beam path, relative to light path in vacuum, due to all disturbances occurred in time ts = ds /c  t.

(m/mo)* - represents the number of individual material particles per unit of volume, referred to a standard material (water at 20ºC), multiplied by the material activity to produce refraction, a*.

The calculation of these parameters is explained ahead.


  1. Φd ; activity of D axis, between water molecules (distance = d0 pm).


Not all positions on D axis are available for light refraction. Material particles decay forming radiation, we design as X Beams. Each X Beam form null particles beams, Z Beams. These e(0) beams are related with energy in X Beam vicinity.  The quantity of these Z Beams at each molecule vicinity depends on the number of active Poles, Po*, that is, Poles where decay reactions occur. Poles not active are involved in individual material particles bonds.  

The number of Z Beams formed at a distance dL ε, from water molecule centre, is given by:


                                                (Po*/16.bW) . dL 2                                                           (4) 


considering that Po* have a regular distribution around material molecules, and only a fraction of 1/8 has interference with light beams, in our case.


 In the light beam, the number of  e(0) formed at a distance l λ, is:


                                                (1/2bL) . λ 2                                                                                                  (5)


If we define the distance  dL as the point in D axis where the number of e(0) formed, in the vicinity of D axis is the same as the e(0) formed in one light wavelength, in same time. Then,


                                                dL  =  λ /(Po*/8)1/2                                                       (6)


not accounting for the relativist effects on time decay in both situations, that is, considering bW  =   bL.        


The value of dL e, gives the limit of the zone around water molecules e(0) concentration in higher than in light beams. At this zone light beams are deviated but the phenomenon is not refraction.


Then, referred two water molecules, the portion of D axis for light refraction activity, d*, is:


                                                d*  =  do – 2. λ / (Po*/8)1/2                                           (7)


and, the axis activity per unit of length, Fd, is:


                                                Fd     =  1 – 2. λ / (do. (Po*/2)1/2)                                   (8)



  1. 2.     Light Photons displacement.

When a light photon crosses the region between two water molecules, an inter-action with null particles from water X Beams (from water molecule decay) can occur.  At Figure 1 we present a light beam, with dS pm length, and λ pm of wavelength, crossing D axis, between two water molecules.

Imagine if, at time 1 Ω, mode (-), light photon is at position (D) at 1 ε of distance from D axis (Figure 1 and 2), and one e(0) particle from water X Beam, is at position (E) at 1 ε of distance from position (C), the cross point of light beam with X Beam.

At these conditions, light photon can permute position with e(0) particle (Figure 2 (a)): 

Zi(+3) . e(0)  ®  e(0) . Zi(+3)                                    (9)


Figure 2


At time 2 Ω, mode (+), the photon is at axis D at position (E). At these conditions photon, Zi(+3) do not react with e(+,+) particles in empty space.

At time 3 Ω, mode (-), another dislocation can occur if at position D, at time 3 Ω, exist one e(0) particle from water molecule X Beam. Note that between material particles, pairs of parallels X Beams are formed, separated by 2 ε of distance.

The result of these interferences of e(0) from water X Beams with light beams, is the retard of light travel in water, compared with in vacuum, represented by Δo pm at Figure 1 (for a standard time ts = ds.c-1 τ). We name disturbance the occurrence of one set of dislocations as presented at Figure 2.

We represent by d the total of decrease of light travel length, during the standard time, due to the total of disturbances, k, as:


                                                δ  =  k . Δo     ε                                      (10)



  1. 3.     Φl. activity of light wavelength λ


In the previous section we described the dislocation of light photons due to the presence of e(0) particles from X Beams. In light refraction, photons dislocate from position (D) to (E) and e(0) particles dislocate in the opposite direction. Following the Second Law of Thermodynamic, this displacement only happens if e(0) concentration is lower at (D) vicinity, compared with (E) vicinity.

From the X Beams, produced by water molecule (A) only one portion, reaches (B) molecule. This portion, referred as pi*, is:

                                                            pi* =   Po*/(4.π.do2)                                           (11)


The concentration of e(0) produced by active Poles from water molecule (A), x position, will decrease along D axis, according to:

                                                [e(0)] =  4.(pi*/bW)/(2.π.x)                                           (12)


(bW is the periodicity of photons formation in water X Beams).

The same for e(0) formed by water molecule (B), in the opposite direction. The minimal value of the sum of these two concentrations will be at D axis, position d = ½.do nm.

At this point e(0) concentration value, is:

                                                [e(0)] ½  =   4. pi*/( bW.π.do)                                         (13)


In order to have an e(0) dislocation to produce light refraction, e(0) concentration at light beam must have a lower concentration than value of equation (13).

The concentration of e(0) formed in a light beam, along one wavelength, at a distance x nm from l beginning, is:


                                                [e(0)]x  = 1/( bL. π. x)                                                  (14)


At a certain value of x, we refer as lL, the value of [e(0)]l  equals [e(0)]1/2. At this situation, we have:


         λL =  do  / pi*     = 4.π.do3 / Po*                          (15)

This means that, in order to have refraction, photons in light beam must be located at a distance higher than lL, from the l beginning (Figure 1). 

Then, the activity of light beam, is:


                                                λ*  =  λ -  λL                                                               (16)


Then the light beams activity, for refraction, per unit of length is:


                                                Φl  =   λ*/λ  =   1 - 4.π.do3 / (Po*.λ)                             (17)


  1. 4.     Standard liquid

We need to calculate the activity of different molecules for light refraction. We chose water at 20º C and normal pressure, and light wave length l = 589,00 nm. At these conditions we refer the water activity, a*, as standard refractometric activity:


                                                            ao* =  1                                                            (18)


  1. 5.     Calculation of molecules active poles Po*, Kp and δ 


Total active Poles, Po*, that is, Poles where decay reactions occur, producing X Beams, are calculated using liquids with known refractometric indexes, at given conditions.

The same for other parameters, Kp and δ.


  1. 6.     Calculation of matter per volume unit, at standard conditions, m/m0


We use water at 20ºC and normal pressure conditions to calculate m0 value.



Examples of calculations are present at Sucropedia/Refraction/Water; Sucropedia/ Refraction/Glucose; Sucropedia/Refraction/Fructose; Sucropedia/Refraction/Sucrose (in preparation).

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